Substituting into formulas

To use a formula, replace each letter with its value, write in the × signs, then work it out in the right order.

A formula is a recipe for a number. Put in the values you know, and it gives you the one you want: a perimeter, a cost, a speed.

To substitute into a formula is to replace each letter with its value, then work out the answer. A number written next to a letter, such as $2l$ , means multiply: $2 \times l$ .

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

A perimeter

P = 2l + 2w

What does $2l$ mean? Write the formula again with the × signs in.

P = 2l + 2w

. What is

P

when

l = 5

and

w = 3

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TryTry it on the board and compare the result with your prediction.

Check your value

Read the working under the rectangle. Where did the × signs come from?

Try it yourselfYour edits are saved in this browser.

Set $l = 5$ and $w = 3$ . Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

Three steps

Here is how to substitute, step by step.

Worked exampleBefore each step, say what you think comes next.

$P = 2l + 2w$ Write the formula first.
$l = 5, \; w = 3$ Write down the values you know.
$P = 2 \times 5 + 2 \times 3$ Replace each letter with its value. Write in the × signs that were hidden in $2l$ and $2w$ .
$P = 10 + 6$ Multiply before adding.
$P = 16 \text{ cm}$ Add, and give the units.

Read Leo's working one line at a time. What does $2l$ mean?

The area of a rectangle is $A = lw$ . Find $A$ when $l = 8$ and $w = 3$ .

A plumber charges $C = 5h + 20$ dollars. Find $C$ when $h = 3$ .

Leo works out a perimeter. Click the line where he goes wrong.

ModifyChange the problem on purpose and see what happens.

Other formulas

Each task loads a new formula. The board shows the substitution, but not the answer: work it out yourself.

Try it yourselfYour edits are saved in this browser.

Find $P = 2l + 2w$ when $l = 2.5$ and $w = 4$ . Set the board to check your substitution.

The volume of a box is $V = lwh$ . Find $V$ when $l = 2$ , $w = 3$ and $h = 4$ .

Average speed is $s = d / t$ (distance ÷ time). Find $s$ in km/h when $d = 120$ km and $t = 2$ hours.

A hire company charges $C = 5(h + 4)$ dollars. Find $C$ when $h = 2$ .

PractiseA few on your own, to check you have it.

Substitute

Find $T = 12n + 6$ when $n = 4$ .

Find $y = 3(x - 2)$ when $x = 7$ .

Find $A = \frac{bh}{2}$ (the area of a triangle) when $b = 6$ and $h = 5$ .

Review: in

V = 3t + 10

, which are the constants?Only 103 and 10

V

and

t

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Rectangles with the same area

Find a rectangle with area 24 cm². Then find another one. How many can you find with whole-number sides?

Stretch: which of your rectangles has the smallest perimeter? Use $P = 2l + 2w$ to check.

Try it yourselfYour edits are saved in this browser.

Need a hint?

A = l × w. Which two whole numbers multiply to 24?

Another hint

Try l = 6. What must w be?

Make a rectangle with an area of 24 cm².

Use the board. This ticks when you get there.

For teachers

Students substitute values into formulas, writing in the hidden × signs and using the order of operations, including decimals, division and brackets (AC9M7A01, node 2). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can replace each letter in a formula with its value, writing in the × signs.
I can work out the result in the right order, including brackets and division.
I can show my substitution as a line of working, with units.

Before class

Do the Modify boards yourself first: they show the substitution but hide the value, so students must finish the arithmetic.

The new move is writing the hidden × signs. The common error is reading 2l with l = 5 as 25 (Leo's mistake). Insist on a line of working with × signs, every time, before any arithmetic.

The Run board shows the full working, so students can compare it with their own line.

Running the lesson

Predict: show of hands. Ask a student who chose 48 how they got it, then write 2l = 2 × l on the board.
Run: have students say the board's working line aloud: 'two times five plus two times three'.
Investigate: model the worked example live. Write the formula, the values, then the substituted line, before any arithmetic.
Modify: before brackets, ask what 5(h + 4) means in words.
Make: collect every rectangle the class found with area 24. Is there a pattern?

When students get stuck

A student writes digits side by side (25 for 2l with l = 5). Ask: What sign is hiding between the 2 and the l? Fix: Write the formula again with every × sign in, then substitute.
A student adds before multiplying (5 × 3 + 20 = 75). Ask: Which do you do first: multiply or add? Fix: Multiply and divide before adding and subtracting, unless brackets say otherwise.

Stretch and support

Stretch: Find P = 2l + 2w and A = lw for l = 4.5 and w = 2. Find a rectangle where P and A are the same number.

Support: Give a template with three lines: formula, substituted (with × signs), answer with units.

Where it leads

Next: finding an unknown, where the answer is known and one letter is not. Substitution is also how every solution is checked, in Equations and in Linear relations (tables from rules).

What each step is for

a perimeter: Committing to a value exposes reading 2l with l = 5 as 25, the digits side by side.
check your value: The board writes every substitution out with the × signs, so the student can compare it with their own working.
three steps: Substituting is three steps: write the formula, replace each letter (with × signs in), then work out in order: multiply before adding.
other formulas: One change at a time: a decimal value, three letters multiplied, a division, and brackets that are worked out first.
substitute: Substituting into new formulas with the order of operations, mixed; the review is naming variables and constants from the last lesson.
rectangles with the same area: Choosing values to hit a target reverses substitution: from the answer back to the letters.

Now you can

Replace each letter in a formula with its value, writing in the × signs.
Work out the result in the right order, including brackets and division.
Show my substitution as a line of working, with units.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson3. Finding an unknownWhen a formula's answer is known but one letter is not, substitute what you know. That leaves an equation; solve it by undoing, then check.

Source: content/maths/algebra/substituting/lesson.md